Let us first prove part (REF ): consider a point \(\big [\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}\big ]\in |\overline{\mathcal {J}}_{g,n}^{\beth }|\) , where \(R\)  is an integral domain which is complete with respect to a rank-1 valuation \(\operatorname{val}_R:R\rightarrow \mathbb {R}_{\ge 0}\cup \lbrace \infty \rbrace \)  and having an algebraically closed residue field \(k\) , and let \((\mathcal {C}\rightarrow \operatorname{Spec}R, \mathcal {L})\)  be the induced family of quasi-stable curves endowed with an admissible line bundle. By (REF ), we have that
\(\operatorname{red}_{\overline{\mathcal {J}}_{g,n}}\big ([\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}]\big )=(\mathcal {C}_s,\mathcal {L}_s)\in \overline{\mathcal {J}}_{g,n}(k)\) 
where \(s\)  is the special point of \(\operatorname{Spec}R\) . On the other hand, by Definition REF , we have that
\(\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}\big ([\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}]\big ):=\big (\Gamma (\mathcal {C}_s), \underline{\deg }(\mathcal {L}_s)\big )\in \overline{J}^{trop}_{g,n},\) 
where \(\Gamma (\mathcal {C}_s)\)  is the quasi-stable tropical curve of type \((g,n)\)  whose underlying graph is \(G(\mathcal {C}_s)\)  and whose metric is the metric \(d_{\mathcal {C}}\)  in (REF ).
In particular, we have that
\(\big (G(\mathcal {C}_s),\underline{\deg }(\mathcal {L}_s)\big )=\big ((\Gamma (\mathcal {C}_s)), \underline{\deg }(\mathcal {L}_s)\big ).\) 
By combining (REF ), (REF ) and (REF ), and recalling the definition of \(J_{(G,D)}\)  from Proposition REF (REF ) and of \(\overline{J}^{trop}_{(G,D)}\)  from (REF ), we conclude that (for any given \((G,D)\in \operatorname{\mathcal {QD}iv}_{g,n}\) )
\(\big [\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}\big ]\in (\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}})^{-1}(\overline{J}^{trop}_{(G,D)})\Leftrightarrow \big [\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}\big ]\in (\operatorname{red}_{\overline{\mathcal {J}}_{g,n}})^{-1}\big (|J_{(G,D)}|\big ).\) 
The last assertion follows from what we already proved together with Remarks REF  and REF .
Part (REF ) follows immediately by combining Remark REF , the definitions (REF ) and Proposition REF .
Let us now prove part (REF ), using the notation already introduced in the proof of parts (REF ) and (REF ). Fix \((G,D)\in \operatorname{\mathcal {QD}iv}_{g,n}\)  such that
\((\mathcal {C}_s,\mathcal {L}_s)\in J_{(G,D)}(k)\) . From the the proof of Proposition REF  (see (REF ) and the deformation-theoretic arguments that precedes it), it follows that we have a canonical identification \(E_{J_{(G,D)},(\mathcal {C}_s,\mathcal {L}_s)}=\mathbb {N}^{E(G(\mathcal {C}_s))}\)  and that the monoid homomorphism appearing in (REF )
\(\mathbb {N}^{E(G(\mathcal {C}_s))}=E_{J_{(G,D)},(\mathcal {C}_s,\mathcal {L}_s)}{\widetilde{\psi }} ((R\setminus R^*)/R^*,.)\) 
is induced by the morphism \(E(G(\mathcal {C}_s))\rightarrow (R\setminus R^*)/R^*\)  that sends \(e\in E(G(\mathcal {C}_s))\)  into \([f_e]\in (R\setminus R^*)/R^*\)  where \(f_e\in R\setminus R^*\)  is such that an étale local equation for \(\mathcal {C}\)  at the node \(n_e\)  is given by \(xy=f_e\) . We deduce that the monoid homomorphism
\(\mathbb {N}^{E(G(\mathcal {C}_s))}=E_{J_{(G,D)},(\mathcal {C}_s,\mathcal {L}_s)}{\widetilde{\psi }} \big ((R\setminus R^*)/R^*,.\big ){\overline{\operatorname{val}}_R} \mathbb {R}_{>0}\cup \lbrace \infty \rbrace \) 
is induced by the metric \(d_{\mathcal {C}}:E(G(\mathcal {C}_s))\rightarrow \mathbb {R}_{>0}\cup \lbrace \infty \rbrace \)  of (REF ). From this, it follows that the isomorphism \(\overline{\Psi }_{\overline{\mathcal {J}}_{g,n}}\)  constructed in (REF ) sends the element \(\operatorname{trop}^{an}_{\overline{\mathcal {J}}_{g,n}}\big ([\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}]\big )\in \overline{\sigma }(G,D)^o/H_{(G,D)}\subset \overline{\Sigma }(\overline{\mathcal {J}}_{g,n})\)  to the tropical curve \(\widetilde{\operatorname{trop}}^{an}_{\overline{\mathcal {J}}_{g,n}}\big ([\operatorname{Spec}R\rightarrow \overline{\mathcal {J}}_{g,n}]\big )\in \overline{J}^{trop}_{g,n}\) .
Part (REF ): the commutativity of the upper triangle is proved in part (REF ), the commutativity of the lower triangle is proved in [1]}, the commutativity of the left square has already been observed in (REF ), and the commutativity of the right square follows from the commutativity of the diagram (REF ) together with the surjectivity of the map \(\operatorname{trop}^{an}_{\overline{\mathcal {J}}_{g,n}}\) .
